Symmetry – Definition, Examples, Practice Problems, FAQs
reviewed by Jo-ann Caballes
Updated on October 2, 2026
Symmetry is when a shape or pattern can be divided, rotated, or moved and still match its original form — the two halves of a heart split down the middle, for instance, are mirror images of each other. This guide covers lines of symmetry, five distinct types of symmetry, and worked examples — solid grounding for students working with elementary math tutors on shapes and geometry.
What Is Symmetry in Math?
Symmetry describes an object made of matching parts — most often two identical halves, but also patterns that match after a rotation or a slide. A shape has symmetry any time part of it can be reflected, rotated, or moved and still line up with the rest of the shape.
The most familiar kind is reflective symmetry, where a shape has identical parts facing one another across an axis — this is what’s usually meant when a shape is called “symmetrical.”
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What Is the Line of Symmetry?
The line of symmetry is an invisible axis dividing the identical halves of a symmetrical shape. It’s also called a mirror line or reflection line — the shape on each side is identical and looks reflected, just like a mirror.
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Types of Lines of Symmetry
There are three common types of lines of symmetry:
- Vertical line of symmetry
- Horizontal line of symmetry
- Diagonal line of symmetry
Vertical Line of Symmetry
A vertical line of symmetry runs from the bottom to the top of a symmetrical shape, splitting it into two congruent left-and-right halves. Many geometric shapes have vertical lines of symmetry, including squares and rectangles.
Horizontal Line of Symmetry
A horizontal line of symmetry runs across a symmetrical shape from left to right, with the shape mirrored above and below the line. Many geometric shapes, including rectangles and squares, have horizontal lines of symmetry too.
Diagonal Line of Symmetry
A diagonal line of symmetry runs from corner to corner, also splitting a shape into two identical halves. Shapes with straight corners, such as squares and rhombuses, can be split with diagonal lines of symmetry. A circle has no corners, so its many lines of symmetry all pass straight through its center instead — they aren’t described as “diagonal” in the corner-to-corner sense.
Types of Symmetry
Beyond simple mirror images, there are several distinct types of symmetry:
- Reflective (reflexive) symmetry
- Translational symmetry
- Rotational symmetry
- Glide symmetry
- Point symmetry
Reflective (Reflexive) Symmetry
Reflective symmetry — also called reflexive symmetry — is where a shape is reflected over a line of symmetry and is identical on both sides. This is the most widely understood type of symmetry and can be seen in many shapes.
Translational Symmetry
Translational symmetry is when an image or figure is moved (or translated) without being altered, reflected, or rotated — if you slid the copies on top of each other, they’d match perfectly. It’s key to understanding patterns, particularly tessellations, where identical shapes fit together with no gaps in between, like the honeycomb below.
Rotational Symmetry
Rotational symmetry requires a central point: rotate an image by less than 360° and it still matches the original shape at a different angle. A snowflake has six-fold rotational symmetry — it matches its original form every 60° as it’s rotated a full 360°.
Glide Symmetry
Glide symmetry (or glide reflection) describes an image that’s been reflected and then slid along a line, ending up identical to the original. A classic example is human footprints walking in a straight line, each one a mirror image of the last, shifted forward.
What Is Point Symmetry?
Point symmetry describes a shape that looks identical to its original when rotated exactly 180° around a central point. Letters like Z, H, N, and O all show point symmetry — rotate any of them a half-turn around their center, and they land right back on themselves. Point symmetry is closely related to the axis of symmetry, since both describe a fixed reference around which the shape is balanced.
Solved Math Tasks: Examples
Use the figure below — a square labeled ABCD with center S — to answer the following questions.
Solved math task 1: Draw all lines of symmetry in the figure and the center of rotational symmetry. The answer is shown in the figure above: a square has 4 lines of symmetry (2 diagonals, 1 vertical, 1 horizontal), all meeting at center S.
Solved math task 2: What different types of symmetry are there in the image? Answer: line symmetry and rotational symmetry — the square has 4 lines of symmetry and also matches itself after every 90° turn.
Solved math task 3: How many angles of rotational symmetry are there? Answer: 4 — at 90°, 180°, 270°, and 360°, matching the square’s four-fold rotational symmetry.
Symmetry shows up everywhere in the natural world, too — from snowflakes to flower petals — which is part of what makes it such an engaging topic to explore through math in nature.
Symmetry Worksheets
Brighterly offers math worksheets on symmetry to help your kid learn and practice:
Frequently Asked Questions About Symmetry
What Is Symmetry in Math?
Symmetry in math describes a shape or pattern made of matching parts — most often two mirror-image halves, but also patterns that match after being rotated or slid into a new position. A shape is symmetrical if part of it can be reflected, rotated, or moved and still line up with the rest.
What Is a Line of Symmetry?
A line of symmetry is an imaginary line that splits a shape into two halves that are exact mirror images of each other — also called a mirror line or reflection line. Shapes can have one line of symmetry, several, or none at all.
What’s the Difference Between Reflective, Rotational, and Translational Symmetry?
Reflective symmetry means a shape matches its mirror image across a line. Rotational symmetry means a shape matches itself after being turned less than a full 360° around a center point. Translational symmetry means a shape or pattern matches itself after being slid to a new position, without any flipping or turning.
How Many Lines of Symmetry Does a Square Have?
A square has 4 lines of symmetry: one vertical, one horizontal, and two diagonal lines running corner to corner. All four lines pass through the square’s center.
Does Every Shape Have a Line of Symmetry?
No. Many irregular shapes have no lines of symmetry at all, since no line can split them into two identical mirror-image halves. A scalene triangle, for example, has no lines of symmetry, while an equilateral triangle has three.
What Is Point Symmetry?
Point symmetry is when a shape looks identical to its original after being rotated exactly 180° around a central point. Letters like Z, H, N, and O all show point symmetry, since rotating any of them a half-turn lands them back on themselves.
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